Metrization of the Gromov-Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Khezeli, Ali^1,2 

[1] Tarbiat Modares Univ, Fac Math Sci, Dept Appl Math, POB 4115-134, Tehran, Iran

[2] Inst Res Fundamental Sci IPM, Sch Math, POB 19395-5746, Tehran, Iran

关键词: Gromov-Hausdorff metric;    Gromov-Hausdorff convergence;    Gromov-Hausdorff-Prokhorov metric;    Boundedly-compact;    Strassen's theorem;   

DOI  :  10.1016/j.spa.2019.11.001

来源: Elsevier

【 摘 要 】

In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov-Hausdorff topology, is a Polish space. The same is done for the Gromov-Hausdorff-Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement to study random boundedly-compact pointed (measured) metric spaces, which is the main motivation of this work. In particular, this provides a unified framework for studying random graphs, random discrete spaces and random length spaces. The proofs use a generalization of the classical theorem of Strassen, presented here, which is of independent interest. This generalization provides an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in terms of approximate couplings. A Strassen-type result is also provided for the Gromov-Hausdorff-Prokhorov metric for compact spaces. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free